Uncertainty principle and geometry of the infinite Grassmann manifold

Abstract

We study the pairs of projections PIf=χIf,QJf=(χJf) ˇ, f∈L^2(R^n), where I,J⊂R^n are sets of finite positive Lebesgue measure, χI,χJ denote the corresponding characteristic functions and ˇ, ˇ denote the Fourier-Plancherel transformation L^2(R^n)→L^2(R^n) and its inverse. These pairs of projections have been widely studied by several authors in connection with the mathematical formulation of Heisenberg´s uncertainty principle. Our study is done from a differential geometric point of view. We apply known results on the Finsler geometry of the Grassmann manifold P(H) of a Hilbert space H to establish that there exists a unique minimal geodesic of P(L^2(R^n)), which is a curve of the δ(t)=e^{itXI,J}P^{Ie−itXI,J} which joins PI and QJ and has length π/2. Here X_I,J is a selfadjoint operator determined by the sets I,J. As a consequence we deduce that if H is the logarithm of the Fourier-Plancherel map, then ∥[H,PI]∥≥π/2. The spectrum of X_I,J is denumerable and symmetric with respect to the origin, and it has a smallest positive eigenvalue γ(X_I,J) which satisfies cos(γ(X_I,J))=∥PIQJ∥.